A372272 -id:A372272 - cp's OEIS frontend

A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

3, 6, 0, 7, 6, 1, 5, 7, 3, 0, 4, 8, 1, 3, 8, 6, 0, 7, 5, 6, 9, 8, 3, 3, 5, 1, 3, 8, 3, 7, 7, 1, 6, 1, 1, 1, 6, 6, 1, 5, 2, 1, 8, 9, 2, 7, 4, 6, 7, 4, 5, 4, 8, 2, 2, 8, 9, 7, 3, 9, 2, 4, 0, 2, 3, 7, 1, 4, 0, 0, 3, 7, 8, 3, 7, 2, 6, 1, 7, 1, 8, 3, 2, 0, 9, 6, 2

Offset: 0

A.H.M. Smeets, Apr 03 2025

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
 k | zeros                     | corresponding weights
---+---------------------------+--------------------------
 2 | A020760                   | A000007*10
 3 | A010513/10                | A010716
 4 | A372267, A372268          | A382103, A382104
 5 | A372269, A372270          | A382105, A382106
 6 | A372271, A372272, A372273 | A382107, this sequence, A382687
 7 | A372274, A372275, A372276 | A382688, A382689, A382690

			0.36076157304813860756983351383771611166152189274674...

A.H.M. Smeets, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=6.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Cf. A372272.

A372267 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.

Original entry on oeis.org

3, 3, 9, 9, 8, 1, 0, 4, 3, 5, 8, 4, 8, 5, 6, 2, 6, 4, 8, 0, 2, 6, 6, 5, 7, 5, 9, 1, 0, 3, 2, 4, 4, 6, 8, 7, 2, 0, 0, 5, 7, 5, 8, 6, 9, 7, 7, 0, 9, 1, 4, 3, 5, 2, 5, 9, 2, 9, 5, 3, 9, 7, 6, 8, 2, 1, 0, 2, 0, 0, 3, 0, 4, 6, 3, 2, 3, 7, 0, 3, 4, 4, 7, 7, 8, 7, 5

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.339981043584856264802665759103244687200575869770914352592953...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 25.4, n=4
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 4.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[4, #] &, 3], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Smallest positive root of 35*x^4 - 30*x^2 + 3 = 0.

Equals sqrt((3-2*sqrt(6/5))/7).

A372268 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.

Original entry on oeis.org

8, 6, 1, 1, 3, 6, 3, 1, 1, 5, 9, 4, 0, 5, 2, 5, 7, 5, 2, 2, 3, 9, 4, 6, 4, 8, 8, 8, 9, 2, 8, 0, 9, 5, 0, 5, 0, 9, 5, 7, 2, 5, 3, 7, 9, 6, 2, 9, 7, 1, 7, 6, 3, 7, 6, 1, 5, 7, 2, 1, 9, 2, 0, 9, 0, 6, 5, 2, 9, 4, 7, 1, 4, 9, 5, 0, 4, 8, 8, 6, 5, 7, 0, 4, 1, 6, 2

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.861136311594052575223946488892809505095725379629717637615721...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=4
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 4.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[4, #] &, 4], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Largest positive root of 35*x^4 - 30*x^2 + 3 = 0.

Equals sqrt((3+2*sqrt(6/5))/7).

A372269 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.

Original entry on oeis.org

5, 3, 8, 4, 6, 9, 3, 1, 0, 1, 0, 5, 6, 8, 3, 0, 9, 1, 0, 3, 6, 3, 1, 4, 4, 2, 0, 7, 0, 0, 2, 0, 8, 8, 0, 4, 9, 6, 7, 2, 8, 6, 6, 0, 6, 9, 0, 5, 5, 5, 9, 9, 5, 6, 2, 0, 2, 2, 3, 1, 6, 2, 7, 0, 5, 9, 4, 7, 1, 1, 8, 5, 3, 6, 7, 7, 5, 5, 2, 9, 1, 0, 3, 5, 8, 0, 3

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.538469310105683091036314420700208804967286606905559956202231...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 4.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[5, #] &, 4], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Smallest positive root of 63*x^4 - 70*x^2 + 15 = 0.

Equals sqrt(5-2*sqrt(10/7))/3.

A372270 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.

Original entry on oeis.org

9, 0, 6, 1, 7, 9, 8, 4, 5, 9, 3, 8, 6, 6, 3, 9, 9, 2, 7, 9, 7, 6, 2, 6, 8, 7, 8, 2, 9, 9, 3, 9, 2, 9, 6, 5, 1, 2, 5, 6, 5, 1, 9, 1, 0, 7, 6, 2, 5, 3, 0, 8, 6, 2, 8, 7, 3, 7, 6, 2, 2, 8, 6, 5, 4, 3, 7, 7, 0, 7, 9, 4, 9, 1, 6, 6, 8, 6, 8, 4, 6, 9, 4, 1, 1, 4, 2

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.906179845938663992797626878299392965125651910762530862873762...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 4.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[5, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Largest positive root of 63*x^4 - 70*x^2 + 15 = 0.

Equals sqrt(5+2*sqrt(10/7))/3.

A372271 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.

Original entry on oeis.org

2, 3, 8, 6, 1, 9, 1, 8, 6, 0, 8, 3, 1, 9, 6, 9, 0, 8, 6, 3, 0, 5, 0, 1, 7, 2, 1, 6, 8, 0, 7, 1, 1, 9, 3, 5, 4, 1, 8, 6, 1, 0, 6, 3, 0, 1, 4, 0, 0, 2, 1, 3, 5, 0, 1, 8, 1, 3, 9, 5, 1, 6, 4, 5, 7, 4, 2, 7, 4, 9, 3, 4, 2, 7, 5, 6, 3, 9, 8, 4, 2, 2, 4, 9, 2, 2, 4

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.238619186083196908630501721680711935418610630140021350181395...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 4], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Smallest positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

Original entry on oeis.org

9, 3, 2, 4, 6, 9, 5, 1, 4, 2, 0, 3, 1, 5, 2, 0, 2, 7, 8, 1, 2, 3, 0, 1, 5, 5, 4, 4, 9, 3, 9, 9, 4, 6, 0, 9, 1, 3, 4, 7, 6, 5, 7, 3, 7, 7, 1, 2, 2, 8, 9, 8, 2, 4, 8, 7, 2, 5, 4, 9, 6, 1, 6, 5, 2, 6, 6, 1, 3, 5, 0, 0, 8, 4, 4, 2, 0, 0, 1, 9, 6, 2, 7, 6, 2, 8, 8

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.932469514203152027812301554493994609134765737712289824872549...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros
  ---+--------------------------
   2 | A020760
   3 | A010513/10
   4 | A372267, A372268
   5 | A372269, A372270
   6 | A372271, A372272, A372273
   7 | A372274, A372275, A372276

First[RealDigits[Root[LegendreP[6, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Largest positive root of 231*x^6 - 315*x^4 + 105*x^2 - 5 = 0.

A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

4, 0, 5, 8, 4, 5, 1, 5, 1, 3, 7, 7, 3, 9, 7, 1, 6, 6, 9, 0, 6, 6, 0, 6, 4, 1, 2, 0, 7, 6, 9, 6, 1, 4, 6, 3, 3, 4, 7, 3, 8, 2, 0, 1, 4, 0, 9, 9, 3, 7, 0, 1, 2, 6, 3, 8, 7, 0, 4, 3, 2, 5, 1, 7, 9, 4, 6, 6, 3, 8, 1, 3, 2, 2, 6, 1, 2, 5, 6, 5, 5, 3, 2, 8, 3, 1, 2

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.405845151377397166906606412076961463347382014099370126387043...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | this sequence, A372275, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.1, 0.5, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Smallest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

7, 4, 1, 5, 3, 1, 1, 8, 5, 5, 9, 9, 3, 9, 4, 4, 3, 9, 8, 6, 3, 8, 6, 4, 7, 7, 3, 2, 8, 0, 7, 8, 8, 4, 0, 7, 0, 7, 4, 1, 4, 7, 6, 4, 7, 1, 4, 1, 3, 9, 0, 2, 6, 0, 1, 1, 9, 9, 5, 5, 3, 5, 1, 9, 6, 7, 4, 2, 9, 8, 7, 4, 6, 7, 2, 1, 8, 0, 5, 1, 3, 7, 9, 2, 8, 2, 6

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.741531185599394439863864773280788407074147647141390260119955...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, this sequence, A372276 | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.6, 0.8, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Middle positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

A372276 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 7.

Original entry on oeis.org

9, 4, 9, 1, 0, 7, 9, 1, 2, 3, 4, 2, 7, 5, 8, 5, 2, 4, 5, 2, 6, 1, 8, 9, 6, 8, 4, 0, 4, 7, 8, 5, 1, 2, 6, 2, 4, 0, 0, 7, 7, 0, 9, 3, 7, 6, 7, 0, 6, 1, 7, 7, 8, 3, 5, 4, 8, 7, 6, 9, 1, 0, 3, 9, 1, 3, 0, 6, 3, 3, 3, 0, 3, 5, 4, 8, 4, 0, 1, 4, 0, 8, 0, 5, 7, 3, 0

Offset: 0

Pontus von Brömssen, Apr 25 2024

			0.949107912342758524526189684047851262400770937670617783548769...

Paolo Xausa, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.
A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.
Eric Weisstein's World of Mathematics, Legendre Polynomial.
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.
Wikipedia, Legendre polynomials.
Index entries for algebraic numbers, degree 6.

Cf. A008316, A100258.
There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
   k | zeros                           | corresponding weights for Legendre-Gauss quadrature
  ---+---------------------------------+----------------------------------------------------
   2 | A020760                         | A000007*10
   3 | A010513/10                      | A010716
   4 | A372267, A372268                | A382103, A382104
   5 | A372269, A372270                | A382105, A382106
   6 | A372271, A372272, A372273       | A382107, A382686, A382687
   7 | A372274, A372275, this sequence | A382688, A382689, A382690

First[RealDigits[Root[LegendreP[7, #] &, 7], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

solve (x = 0.8, 1.0, 429*x^6 - 693*x^4 + 315*x^ - 35) \\ A.H.M. Smeets, May 31 2025

Largest positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

cp's OEIS Frontend

A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

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A372267 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.

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A372268 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.

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A372269 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 5.

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A372270 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.

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A372271 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 6.

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A372273 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 6.

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Mathematica

Formula

A372274 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 7.

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